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Weir discharge is provided by the general formula:

Formula: weirs discharge

where :

Q = flow rate, in m3·s–1 (or L·s–1),

μ = weir discharge coefficient,

Ls = weir crest width, in m,

h = water sheet height, in m (or cm),

g = gravity acceleration in m·s–2, (9.81 in Paris).

Additionally, P is used for the «height of the weir crest above the upstream channel» and L for the canal width upstream from the weir.

thin-walled rectangular weir with a low approach velocity

Formula: weirs - Thin-walled rectangular weir with a low approach velocity

in the case of a reservoir outlet for instance.

special case of a circular overflow weir

for an overflow having a diameter of 0.20 m < Ø < 0.70 m with enough nappe fall to avoid any downstream reaction.

Formula: weirs  - Special case of a circular overflow weir

thin-walled rectangular weir on a channel

weir with suppressed and contractions (Ls = L), with free nappe flow (figure 43)

This definition applies to a weir when the crest thickness e is less than half of load h, when flow is such that, at atmospheric pressure, it leaves an air-filled space between the water sheet and the crest’s downstream wall, and when the overflow blade width is exactly the same as that of the channel.

End contraction weir Secured image
Figure 43. End contraction weir

The flow rate coefficient µ is provided by one of the following formulae :

  • Bazin formula (1898), generally used in France :
Formula: weirs  - Bazin formula
  • Formula suggested by the Société des Ingénieurs et Architectes Suisses (Society of Swiss Engineers and Architects - SIA):
Formula: weirs - Formula "Société des Ingénieurs et Architectes Suisses"

These formulae, where h and P are expressed in m, can be used for heads on the weir crest h of between 0.10 and 0.60 m 9 in the case of the Bazin formula, and between 0.025 m and 0.80 m in the case of the SIA formula, the latter providing slightly inferior results to those of the Bazin formula.

The following conditions need to be verified:

  • for Bazin: P between 0.20 et 2 m.
  • for SIA: P greater than h.

Finally, h must be measured at a threshold distance equal to at least five times the maximum head on the weir crest. When there is not enough aeration below the nappe (depressed nappe), the flow rate increases and its law becomes poorly defined; this is not acceptable for a measuring weir.

end contraction weir

The SIA puts forward the following formula for µ :

Formula: weirs  - End contraction weir

The simplified Francis formula should also be noted :

Formula: weirs - simplified Francis formula

where the excress width on either side of the crest must be equal to at least 3 h, the height measured at least 2 m upstream from the weir crest.

Flow rate length BazinSecured image
Table 72. Flow rate in L · s–1 per m of crest length according to Bazin(1)
Flow rate length SIASecured image
Table 73. Flow rate in L · s–1 per m of crest length according to SIA(1)

thin-walled tringular weir

Formula: Thin-walled triangular weir

where :

Q = flow rate, in m3·s–1,

μ = Bazin flow rate coefficient for a thin wall, rectangular weir with suppressed end concentrations,

h = height on the weir crest, in m,

θ =angle at the weir apex.

The flow rate for a triangular weir can be deduced from the flow rate for a rectangular weir with suppressed end contractions, for the same height on the weir crest and the same height of weir crest above floor of channel, multiplying this flow rate by:

Formula: Thin-walled triangular weir - rate

For θ = 90°, the Thomson formula is sometimes used:

Formula: Thin-walled triangular weir - Thomson formula

This is a very approximate formula because it does not allow for the effect of the height of the weir crest above the floor of the channel.